Orthogonal Matrices - Examples with Solutions

\( \) \( \) \( \)

Definition of Orthogonal Matrices

An \( n \times n \) matrix whose columns form an orthonormal set is called an orthogonal matrix.
As a reminder, a set of vectors is orthonormal if each vector is a unit vector ( length or norm of the vector is equal to \( 1\)) and each vector in the set is orthogonal to all other vectors in the set.
These are examples of orthogonal matrices.
a) \( \begin{bmatrix} 1 & 0 & 0\\ 0 & 0 & 1 \\ 0 & -1 & 0\\ \end{bmatrix} \)        b) \( \begin{bmatrix} \dfrac{1}{\sqrt {10}} & 0 & - \dfrac{3}{\sqrt {10}}\\ \dfrac{3}{\sqrt {10}} & 0 & \dfrac{1}{\sqrt {10}} \\ 0 & -1 & 0\\ \end{bmatrix} \)        c) \( \begin{bmatrix} \dfrac{1}{\sqrt{6}}&0&\dfrac{5}{\sqrt{30}}\\ \:\dfrac{1}{\sqrt{6}}&-\dfrac{2}{\sqrt{5}}&-\dfrac{1}{\sqrt{30}}\\ \:\dfrac{2}{\sqrt{6}}&\dfrac{1}{\sqrt{5}}&-\dfrac{2}{\sqrt{30}} \end{bmatrix} \)        d) \( \begin{bmatrix} \cos x& -\sin x \\ \sin x&\cos x \end{bmatrix} \)



Properties of Orthogonal Matrices

A list of the most important properties of orthogonal matrices is given below. If \( Q \) is an orthogonal matrix, then

  1.   \( Q^{-1} = Q^T \) ; this the most important property of orthogonal matrices as the inverse is simply the transpose.
  2.   the rows of \( Q \) form an orthonormal set.
  3.   \( Q^{-1} \) is an orthogonal matrix
  4.   det \( ( Q ) = \pm 1 \)
  5.   if \( \lambda \) is an eigrnvalue of Q, then \( |\lambda| = 1 \)
  6.   if \( Q_1 \) and \( Q_2 \) are \( n \times n \) orthogonal matrices, then \( Q_1 Q_2 \) is also an orthogonal matrix.



Examples with Solutions

Example 1
The matrices \( Q_1 = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \) and \( Q_2 = \begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix} \) are orthogonal. Verify that the product \( Q_1 Q_2 \) is also orthogonal (Property 6 above)

Solution
We first calculate the product \( Q_1 Q_2 \)
\( Q_1 Q_2 = \begin{bmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & -1 \end{bmatrix} = \begin{bmatrix} 0&0&-1\\ 0&-1&0\\ 1&0&0 \end{bmatrix} \)
Let \( \textbf v_1 , \textbf v_2 , \textbf v_3 \) be the columns of the matrix \( Q_1 Q_2 \) found above.
\( \textbf {v}_1 = \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix} \) , \( \textbf {v}_2 = \begin{bmatrix} 0\\ -1\\ 0 \end{bmatrix} \) , \( \textbf {v}_3 = \begin{bmatrix} -1\\ 0\\ 0 \end{bmatrix} \)
Let us calculate the length or norm of each column
\( || \textbf {v}_1 || = \sqrt {0^2+0^2+1^2} = 1 \)
\( || \textbf {v}_2 || = \sqrt {0^2+(-1)^2+0^2} = 1 \)
\( || \textbf {v}_3 || = \sqrt {(-1)^2+0^2+0^2} = 1 \)
All three vectors are unit vectors.
Calculate the inner product of all pairs of vectors that can be made from the vectors \( \textbf v_1 , \textbf v_2 , \textbf v_3 \)
\( \textbf {v}_1 \cdot \textbf {v}_2 = 0 \cdot 0 + 0 \cdot (-1) + 1 \cdot 0 = 0 \)
\( \textbf {v}_1 \cdot \textbf {v}_3 = 0 \cdot (-1) + 0 \cdot 0 + 1 \cdot 0 = 0 \)
\( \textbf {v}_2 \cdot \textbf {v}_3 = 0 \cdot (-1) + (-1) \cdot 0 + 0 \cdot 0 = 0 \)
The three vectors form an orthogonal set.
The three columns of the matrix \( Q_1 Q_2 \) are orthogonal and have norm or length equal to 1 and are therefore orthonormal.



Example 2
Use a calculator to find the inverse of the orthogonal matrix matrix \( Q = \begin{bmatrix} 0 & 0 & 1 \\ -1 & 0 & 0 \\ 0 & -1 & 0 \end{bmatrix} \) and verify Property 1 above.

Solution
Use any matrix calculator to find
\( Q^{-1} = \begin{bmatrix} 0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0 \end{bmatrix} \)
Find the transpose of matrix \( Q \)
\( Q^T = \begin{bmatrix} 0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0 \end{bmatrix} \)
Hence \( Q^{-1} = Q^T \) , property 1 above.



Example 3
Find the real constants \( a \) and \( b \) in the matrix \( Q = \begin{bmatrix} -1 & 0 & 0 \\ 0 & \dfrac{1}{\sqrt 2} & a \\ 0 & \dfrac{1}{\sqrt 2} & b \end{bmatrix} \) such that \( Q \) is orthogonal.

Solution
Let \( \textbf v_1 , \textbf v_2 , \textbf v_3 \) be the columns of the matrix \( Q \) given above such that
\( \textbf {v}_1 = \begin{bmatrix} -1\\ 0\\ 0 \end{bmatrix} \) , \( \textbf {v}_2 = \begin{bmatrix} 0\\ \dfrac{1}{\sqrt 2}\\ \dfrac{1}{\sqrt 2} \end{bmatrix} \) , \( \textbf {v}_3 = \begin{bmatrix} 0\\ a\\ b \end{bmatrix} \)
Two sets of conditions for matrix \( Q \) to be orthogonal:
1) The norm of each column \( \textbf v_1 , \textbf v_2 , \textbf v_3 \) must equal to 1
\( || \textbf v_1 || = \sqrt {(-1)^2+0^2+0^2} = 1 \)
\( || \textbf v_2 || = \sqrt {0^2+ \left(\dfrac{1}{\sqrt 2} \right)^2+\left(\dfrac{1}{\sqrt 2} \right)^2 } = 1 \)
\( || \textbf v_3 || = \sqrt {0^2+a^2+b^2} = \sqrt {a^2+b^2} = 1 \)         (I)
2) The inner product of any two vectors must be equal to zero (orthogonal vetcors)
\( \textbf {v}_1 \cdot \textbf {v}_2 = (-1) \cdot 0 + 0 \cdot \left(\dfrac{1}{\sqrt 2} \right) + 0 \cdot \left(\dfrac{1}{\sqrt 2} \right) = 0 \)
\( \textbf {v}_1 \cdot \textbf {v}_3 = (-1) \cdot 0 + 0 \cdot a + 0 \cdot b = 0 \)
\( \textbf {v}_2 \cdot \textbf {v}_3 = 0 \cdot 0 + \dfrac{1}{\sqrt 2} \cdot a + \dfrac{1}{\sqrt 2} \cdot b = \dfrac{1}{\sqrt 2} \cdot a + \dfrac{1}{\sqrt 2} \cdot b = 0 \)         (II)
For all conditions to be satisfied, we need to solve equations (I) and (II) above
Equation (II) above gives \( a = - b \)
Substitute \( a \) by \( - b \) in equation (I) to obtain
\( \sqrt {2 b^2 } = 1 \)
Solve to obtain
\( b = \pm \dfrac{1}{\sqrt 2} \)
Two solutions to the above question
\( a = - \dfrac{1}{\sqrt 2} \) and \( b = \dfrac{1}{\sqrt 2} \)
\( a = \dfrac{1}{\sqrt 2} \) and \( b = - \dfrac{1}{\sqrt 2} \)



Questions (with solutions given below)

  • Part 1
    Matrix \( Q = \begin{bmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} \) is orthogonal. Verify the first 5 properties, listed above, for matrix \( Q \).
  • Part 2
    1) Which of the following matrices are orthogonal?
    a) \( A = \begin{bmatrix} \dfrac{1}{\sqrt 2} & \dfrac{1}{\sqrt 2} \\ - \dfrac{1}{\sqrt 2} & \dfrac{1}{\sqrt 2} \end{bmatrix} \) , b) \( B = \begin{bmatrix} -1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{bmatrix} \) , c) \( C = \begin{bmatrix} \dfrac{1}{\sqrt {10}} & - \dfrac{3}{\sqrt {10}} & 0 \\ 0 & 0 & 1 \\ \dfrac{3}{\sqrt {10}} & \dfrac{1}{\sqrt {10}} & 0 \end{bmatrix} \)
  • Part 3
    1) Find all matrices of the form \( A = \begin{bmatrix} p & q\\ \dfrac{1}{\sqrt {3}} & r \end{bmatrix} \) that are orthogonal.
  • Part 4
    Find the inverse of matrix \( A = \begin{bmatrix} 0 & \dfrac{5}{3 \sqrt 5} & \dfrac{2}{3} \\ \dfrac{1}{\sqrt 5} & - \dfrac{4}{3 \sqrt 5} & \dfrac{2}{3} \\ \dfrac{2}{\sqrt 5} & \dfrac{2}{3 \sqrt 5} & - \dfrac{1}{3} \end{bmatrix} \)



Solutions to the Above Questions

More References and links